The expression is $\sqrt{x^2 - 4x + 3}$.

To simplify it, let's first factor the quadratic expression under the square root.

Step 1: Factor the quadratic expression

We need to factor $x^2 - 4x + 3$.

$x^2 - 4x + 3 = (x - 3)(x - 1)$

So, the original expression becomes:

$\sqrt{(x - 3)(x - 1)}$

Step 2: Consider domain restrictions

For the square root function to be defined, the expression inside the square root must be non-negative. Therefore, we need:

$(x - 3)(x - 1) \geq 0$

Now, solve the inequality by determining when the product is non-negative.

• The zeros are at $x = 1$ and $x = 3$.
• Use a number line to test intervals: $(-\infty, 1)$, $(1, 3)$, and $(3, \infty)$.

The product is positive in the intervals $(-\infty, 1] \cup [3, \infty)$.

Final result:

$\sqrt{x^2 - 4x + 3} = \sqrt{(x - 3)(x - 1)} \text{ for } x \in (-\infty, 1] \cup [3, \infty)$

Would you like further details or any clarifications?

Here are 5 related questions:

1. How do you solve inequalities involving quadratic expressions?
2. Can all quadratic expressions be factored easily?
3. What happens if the expression inside the square root is negative?
4. How do you determine the domain of a function with square roots?
5. What are the different methods to factor quadratic equations?

Tip: Always check the domain of square root functions to avoid undefined values.